Solids Band Theory Assignment Help

Solids Band Theory

According to Bohr’s theory of atomic spectra and the concept of electronic configuration, the electrons in an isolated atom have certain definite discrete amounts of energy corresponding to different shells and subshells. It means there are well defined energy levels of electrons in an isolated atom. If large numbers of atoms are brought close to one another to form a crystal, they begin to influence each other. Due to this interatomic interaction, there is no appreciable modification in energy levels of electrons in the inner shells but there is a considerable modification in the case of energy levels of the electrons in the outer shells. This is due to the fact that the valence electrons are shared by more than one atom in the crystal.

To understand this modification of energy levels, consider single silicon (Si) or Germanium (Ge) crystal containing N atoms. Each atom in the crystal is situated at a lattice site. For silicon atom, atomic number 14, the electronic configuration is 1s2 2s2 2p6 3s2 3p2and Germanium atom, atomic number 32, the electronic configuration is 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p2. Both the atoms have four valence electrons, i.e. number of electrons in the outermost orbit is 4 (2, s-electrons and 2, p-electrons). Therefore, the total number of valence electrons in the crystal of Si or Ge is 4 N. the maximum number of electrons in the outer orbit of silicon atom can be 8 (= 2s electrons + 6p electrons). It means for the 4N valence electrons there are 8N available energy states.

The process of splitting of energy levels for Si can be understood by considering the different situations as discussed below:

(i) If the interatomic spacing of the Si atoms is very large (i.e.r = d >> a), there is no interatomic interaction. Each atom in the interaction behaves as a free atom. In this situation each of N atoms has its own identical energy levels, which are sharp, discrete and distinct. The electronic configuration of silicon of atomic number 14 is 1s2 2s2 2p6 3s2 3p2. This shows that the outer two subshells of silicon atom contain two electrons in 3s subshell and two electrons in 3p subshell whereas six electrons are required to completely fill 3p subshell. Thus in the silicon crystal under study, these are 2N electrons completely filling2N possible 3s levels, all of which are of the same energy. There are 6N possible 3p levels, out of which only 2N levels are filled and all the filled 3p levels have the same energy. All these facts follow from Pauli’s exclusion principle.

(ii) When the interatomic spacing r is less than d but greater than c (i.e. c < r < d), there is no visible splitting of energy levels.

(iii) When the interatomic spacing r is equal to c, the interaction between outermost shell electrons (3s2and 3p2) of neighbouring silicon atoms becomes appreciable. As a result, the energy of silicon corresponding to 3s and 3p levels of each atom starts changing i.e. the splitting of these energy levels commences whereas there is no change in the energy of electrons in the inner shells.

(iv) When the interatomic spacing r lies in between b and c (i.e. b < r < c), the energy of electrons corresponding to 3s levels of each atom gets slightly changed. Instead of a single 3s or 3p level, we get a large number of closely packed levels (2N levels corresponding to a single 3s level and 6N levels for a single3p level of an isolated atom). This spreading of energy corresponding to 3s levels reduces the energy gap that existed between 3s and 3p levels of free atom. Since number N is very large (≈1029 atom/m3) and the energy of 3s and 3p levels is of the order of few electron volt, the levels due to spreading of the energy of 3s or 3p levels are very closely spaced. This collection of closely spaced levels is called an energy band.

(v) When an interatomic spacing r becomes equal to b but greater than a (i.e. r = b > a), the energy gap between 3s and 3p levels completely disappears and the 8N energy levels are continuously distributed. In such a situation, it is not possible to distinguish between the electrons belonging to 3s and 3p subshells. We can only say that 4N are filled and 4N levels are empty.

(vi) When the interatomic spacing r becomes equal to a (i.e. r = a), the actual spacing in the crystal), then at absolute zero, the band of 4N unfilled energy levels by an energy gap called energy band gap, which is denoted by Eg. The lower completely filled band is called valence gap (V.B.) and the upper unfilled band is called conduction band (C.B.) the positions of energy bands in a semiconductor at 0 K is the lowest energy level in the conduction band and is shown as Ec and highest energy level in the valence band is shown asEv. The separation between top of valence band and bottom of conduction band is called energy band gap (energy gap Eg).

Quantum states in silicon crystal having N atoms are shown in the following table: